Eyring Calculator

The Eyring equation k = (kB * T / h) * e^(-ΔG‡ / (R * T)) from transition state theory does not require a reaction-specific pre-exponential factor – the term kB * T / h is a universal constant. This makes it possible to estimate the temperature at which a reaction runs within a practical timeframe solely from the free enthalpy of activation ΔG‡. Conversely, you can determine the enthalpy of activation from a measured rate constant k at a known temperature. This principle is also what lies behind the common rule of thumb "a 10 °C increase doubles the reaction rate".

Enter Values

Do you have A and Ea from an Arrhenius plot instead? Use our Arrhenius Calculator.

⚠️ ΔG‡ here represents the free enthalpy of activation (the energy barrier between reactants and the transition state) – not to be confused with the free enthalpy of reaction ΔG from the general Gibbs energy calculator, which describes the difference between reactants and products.

1/
Rate constant. The time base (1/s, 1/min, ...) is required for the additionally displayed half-life – for reaction orders other than 1st order, this should only be understood as a rough guide.
Absolute temperature at which the reaction takes place.
Free enthalpy of activation ΔG‡ (always molar, e.g., 75 kJ/mol) – the energy barrier to the transition state, NOT the reaction ΔG.

Explanation: The Eyring Equation

What is the Eyring Equation?

The Eyring equation originates from transition state theory and describes how fast a reaction proceeds, based on the energy barrier between the reactants and the unstable transition state.

The major advantage over the Arrhenius equation: Instead of a reaction-specific, often unknown pre-exponential factor A, Eyring uses the term kB * T / h – a combination of fundamental physical constants that is identical for EVERY reaction.

Basic Formula

$$ k = \dfrac{k_B \cdot T}{h} \cdot e^{-\Delta G^\ddagger / (R \cdot T)} $$

Important: ΔG‡ is the Activation Energy, not the Reaction ΔG

ΔG‡ (also written as ΔG≠ or free enthalpy of activation) describes the energy barrier that reactants must overcome on their way to the transition state. This is fundamentally different from the reaction ΔG found in general Gibbs energy calculators, which describes the energetic difference between reactants and products and thus only tells you WHERE a reaction is going (thermodynamics) – not HOW FAST (kinetics). A reaction can have a highly negative reaction ΔG (thermodynamically very favorable) and still proceed extremely slowly if ΔG‡ is large.

Overview of the Three Variables

Rate Constant (k)

k describes how fast a reaction proceeds at a given temperature. Its unit depends on the reaction order (e.g., 1/s for a 1st order reaction).

Free Enthalpy of Activation (ΔG‡)

ΔG‡ is the energy barrier between the reactants and the transition state. The smaller ΔG‡ is, the faster the reaction proceeds at a given temperature.

Temperature (T)

T is the absolute temperature in Kelvin at which the reaction takes place.

The Physical Constants Used (kB, h, R)

kB = 1.380649 * 10⁻²³ J/K (Boltzmann constant), h = 6.62607015 * 10⁻³⁴ J·s (Planck constant), R = 8.314 J/(mol·K) (universal gas constant). All three are fixed physical constants that the calculator uses automatically.

Why this is incredibly practical in the lab

Since the pre-exponential term kB * T / h is the same for all reactions, the activation energy ΔG‡ alone is enough to estimate the temperature at which a reaction will run in a reasonable timeframe – without ever having to determine A experimentally. This makes the Eyring calculator a quick and easy tool for rule-of-thumb estimates in daily lab work.

Solving the Formula for Each Variable

Depending on which variable is requested, the formula is rearranged accordingly:

Solving for ΔG‡

$ \Delta G^\ddagger = -R \cdot T \cdot \ln\left(\dfrac{k \cdot h}{k_B \cdot T}\right) $
Used when k and T are known.

Solving for T

T is present in both the pre-exponential factor and the exponent – a simple algebraic rearrangement is not possible (it requires the Lambert W function). The calculator solves this reliably using numerical methods.

Connection to the RGT Rule (10 °C Doubles the Speed)

The well-known rule of thumb stating that a 10 °C increase in temperature roughly doubles or triples the reaction rate follows directly from this formula: For typical activation enthalpies in the range of 50–100 kJ/mol at around room temperature, this exact factor is achieved. With this calculator, you can calculate this precisely for your specific activation enthalpy instead of relying on the rough rule of thumb.

Example Problems

Two examples demonstrate how to calculate depending on the variable you are looking for.

Example 1: Calculating k

A reaction has ΔG‡ = 75 kJ/mol at T = 298 K. What is the rate constant k?

Given

ΔG‡ = 75000 J/mol, T = 298 K

Solution

k = (kB * T / h) * e^(-ΔG‡ / (R * T))

Pre-exponential factor: kB * 298 / h ≈ 6.21 * 10¹² 1/s. Exponent: -75000 / (8.314 * 298) ≈ -30.28. k ≈ 6.21 * 10¹² * e^(-30.28) ≈ 4.4 * 10⁻¹ 1/s

k ≈ 0.44 1/s

Example 2: Required Temperature for a Practical Reaction Time

A reaction with ΔG‡ = 100 kJ/mol needs to run at k ≈ 1 * 10⁻³ 1/s (reasonably fast, measurable within minutes). At what temperature does this happen?

Given

ΔG‡ = 100000 J/mol, k = 0.001 1/s

Solution

Numerical solution of k = (kB * T / h) * e^(-ΔG‡ / (R * T))

T ≈ 329.8 K ≈ 56.7 °C – the reaction would therefore need a moderately elevated temperature (e.g., a water bath) to proceed within a practical timeframe.

Tips and Common Mistakes

Common Pitfalls

Confusing ΔG‡ with the Reaction ΔG

The free enthalpy of activation ΔG‡ describes the kinetics (how fast), while the reaction ΔG describes the thermodynamics (how far). The two are not directly related – a reaction can be energetically highly favorable (highly negative ΔG) but still remain extremely slow (large ΔG‡).

Using Temperature in °C Instead of Kelvin

The formula always requires the absolute temperature in Kelvin, not degrees Celsius. Don't forget to add 273.15.

Where is this used?

The Eyring equation is particularly convenient when you don't know A or prefer not to determine it experimentally:

  • Quickly estimating the temperature required for a synthesis to run within a practical timeframe
  • Determining ΔG‡ (and subsequently ΔH‡, ΔS‡) from temperature-dependent rate measurements (Eyring plot)
  • Comparing the reactivity of different compounds based on their activation barriers

Frequently Asked Questions about the Eyring Equation

The Eyring equation $ k = \dfrac{k_B \cdot T}{h} \cdot e^{-\Delta G^\ddagger / (R \cdot T)} $ describes how the rate constant k of a reaction depends on temperature T and the Gibbs free energy of activation ΔG‡. Unlike the Arrhenius equation, it does not require a reaction-specific pre-exponential factor A, but instead uses the fundamental physical constants kB (Boltzmann constant) and h (Planck constant). (In its complete form, a transmission coefficient κ is also placed before the term, which is usually assumed to be κ ≈ 1 and is therefore often omitted.)

ΔG‡ (Gibbs free energy of activation) is the energy barrier between reactants and the transition state, determining how fast a reaction proceeds (kinetics). In contrast, the overall reaction ΔG describes the energetic difference between reactants and products, indicating only where a reaction goes (thermodynamics). A reaction can have a strongly negative reaction ΔG and still proceed very slowly if ΔG‡ is large.

In the Arrhenius equation, A is a reaction-specific factor that typically needs to be determined experimentally. In the Eyring equation, the term kB·T/h takes on this role – it consists purely of fundamental physical constants and temperature, making it identical for every reaction. As a result, ΔG‡ alone is sufficient to estimate k without having to measure A beforehand.

T appears in the Eyring equation both in the pre-factor kB·T/h and in the exponent −ΔG‡/(R·T). Consequently, a closed-form algebraic solution for T is not possible (except via the Lambert W function). The calculator therefore solves for T numerically to find the matching temperature for given values of k and ΔG‡.

The well-known rule of thumb that a 10 °C increase approximately doubles to triples the reaction rate follows directly from the Eyring equation: for typical activation free energies of 50–100 kJ/mol around room temperature, the calculated factor falls right into this range. Using the calculator, this effect can be estimated precisely for a specific activation free energy rather than using a blanket estimate.

The calculator uses kB = 1.380649·10⁻²³ J/K (Boltzmann constant), h = 6.62607015·10⁻³⁴ J·s (Planck constant), and R = 8.314 J/(mol·K) (universal gas constant). All three are fixed, experimentally exact physical constants and do not need to be entered manually.